The Maltsev product $\mathcal{V}\circ \mathcal{W}$ of varieties $\mathcal{V}$ and $\mathcal{W}$ of a type $\Sigma$ is a class that consists of all algebras $A$ of the type $\Sigma$, which have a congruence $\theta$ such that the quotient $A/\theta$ belongs to $\mathcal{W}$ and each congruence class of $\theta$ which is a subalgebra of $A$ belongs to $\mathcal{V}$. The class $\mathcal{V}\circ \mathcal{W}$ is closed under subalgebras, direct products, and isomorphic images. However it may not be closed under homomorphic images, and so it may not be a variety. I will present a new sufficient condition for $\mathcal{V}\circ \mathcal{W}$ to be a variety and I will discuss various cases in which this sufficient condition is satisfied.

Sufficient conditions for a Maltsev product of two varieties to be a variety

Tue, Oct. 10 2:30pm (MATH 3…

Lie Theory

Paige Robertson (CU)

X

We will discuss a subfamily of the binary triangle family with weakly increasing columns. This subfamily has a natural poset structure through which we show that the maximal elements are counted by the Fibonacci numbers. We then examine those maximum elements which also have a maximum total sum, or weight, and show that this subset, and its corresponding maximum weights, are enumerated by known OEIS sequences. We conclude with a discussion of two methods of constructing these maximum elements with maximum weight, one recursive and one explicit.

In non-equivariant homotopy theory, formal groups play an essential role. These algebraic objects arise as the E cohomology of infinite complex projective space for E a complex oriented spectrum. If A is a finite abelian group, then there is a notion of an A - equivariant formal group law, and these objects arise naturally as the E cohomology of an A - equivariant analogue of infinite complex projective space. The purpose of this talk is to introduce the audience to these mysterious algebraic objects, and survey some important and recent results in this area. In particular, we will provide a direct proof (as opposed to the beautiful "global homotopy theoretic" proof of Hausmann) that the Hopf algebroid associated with the A - spectrum $M\phantom{\rule{0}{0ex}}{U}_{A}\wedge M\phantom{\rule{0}{0ex}}{U}_{A}$ represents isomorphisms of A-equivariant formal group laws.